Topological Integer Additive Set-Valued Graphs: A Review

Overview & Synopsis

Let í µí± denote a set of non-negative integers and í µí± 0 (í µí±) be the collection of all non-empty subsets of í µí±. An integer additive set-labeling (IASL) of a graph í µí°º is an injective set-valued function í µí±: í µí±(í µí°º) → í µí± 0 (í µí±) where induced function í µí± + : í µí°¸(í µí°º) → í µí± 0 (í µí±) is defined byí µí± + (í µí±¢í µí±£) = í µí±(í µí±¢) + í µí±(í µí±£), where í µí±(í µí±¢) + í µí±(í µí±£) is the sumset of í µí±(í µí±¢) and í µí±(í µí±£). A set-labeling í µí±: í µí±(í µí°º) → í µí± 0 (í µí±) is said to be a topological set-labeling if í µí±(í µí±(í µí°º)) ∪ {∅} is a topology on the ground set í µí± and a set-labeling í µí±: í µí±(í µí°º) → í µí± 0 (í µí±) is said to be a topogenic set-labeling if í µí±(í µí±(í µí°º)) ∪ í µí± + (í µí°¸(í µí°º)) ∪ {∅} is a topology on í µí±. In this article, we critically review some interesting studies on the properties and characteristics of different topological and topogenic integer additive set-labeling of certain graphs.